Finite-dimensional subalgebras of the Virasoro algebra (Q2259162)

Let \(W\) be the Witt algebra of derivations of the complex Laurent polynomial algebra \(C[t,t^{(-1)}]\). \(W\) has a basis \(\{L_n| n \text{an integer}\}\), \(L_n=-(t^n)D\), where \(D=t(d/dt)\), with multiplication \([L_m,L_n]=(m-n)L_{(m+n)}\). \textit{S.-H. Ng} and the reviewer showed that any finite-dimensional subalgebra of \(W\) has dimension at most 3 [J. Pure Appl. Algebra 151, No. 1, 67--88 (2000; Zbl 0971.17008)]. The 3-dimensional ones have bases \(\{L_n,L_0,L_{(-n)}\}\) for each integer \(n>0\). The 1-dimensional ones are of the form \(C^x\) for each non-zero \(x\) in \(W\). For each non-zero \(n\), \(\{L_0,L_n\}\) is the basis of a 2-dimensional subalgebra. However, it is known that there are other 2-dimensional subalgebras not of this form, e.g., see \textit{Y. Su} and \textit{K. Zhao} [J. Algebra 252, No. 1, 1--19 (2002; Zbl 1035.17036)], Lemma 3.2. In the paper under review, the author classifies all the 2-dimensional subalgebras of \(W\). There are two distinct classes. One is the set of subalgebras with basis \(\{D,(t^m)D\}\) for each non-zero integer \(m\). The other class depends on four parameters, and is too technical to describe here. Using this result, the author also classifies the finite-dimensional subalgebras of the Virasoro algebra \(V=W+CK\), \(K\) central, with multiplication \([L_m,L_n]=(m-n)L_{(m+n)}\) if \(m\) is not equal to \(-n\) and \[ [L_m, L_n]= (m-n)L_{(m+n)}+(1/12)((m^2)-m)K \] if \(m=(-n)\). They have dimension at most 4.